Problem 38 - Algebra 2

Question

GEOGRAPHY For Exercises 37 and 38 , use the following information. A Mercator projection map uses a flat projection of Earth in which the distance between the lines of latitude increases with their distance from the equator. The calculation of the location of a point on this projection uses the expression tan \(\left(45^{\circ}+\frac{L}{2}\right)\) where \(L\) is the latitude of the point. Find the exact value of the expression if \(L=60^{\circ} .\)

Step-by-Step Solution

Answer

The exact value is \( 2 + \sqrt{3} \).

Step 1: Understand the Expression

The expression to calculate is \( \tan\left(45^{\circ} + \frac{L}{2}\right) \). Here, \( L \) is the given latitude, which is \( 60^{\circ} \). The goal is to find the exact value of this expression when \( L = 60^{\circ} \).

Step 2: Substitute the Value of L

Substitute \( L = 60^{\circ} \) into the expression. This gives us:\[\tan\left(45^{\circ} + \frac{60^{\circ}}{2}\right) = \tan\left(45^{\circ} + 30^{\circ}\right)\]

Step 3: Simplify the Angle

Compute \( 45^{\circ} + 30^{\circ} \) to simplify the angle inside the tangent function:\[45^{\circ} + 30^{\circ} = 75^{\circ}\]Therefore, the expression becomes \( \tan(75^{\circ}) \).

Step 4: Use Trigonometry to Find Exact Value

To find the exact value of \( \tan(75^{\circ}) \), use the identity for tangent of sum:\[\tan(75^{\circ}) = \tan(45^{\circ} + 30^{\circ}) = \frac{\tan(45^{\circ}) + \tan(30^{\circ})}{1 - \tan(45^{\circ})\tan(30^{\circ})}\]We know \( \tan(45^{\circ}) = 1 \) and \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \).

Step 5: Compute the Final Value

Substitute \( \tan(45^{\circ}) = 1 \) and \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \) into the identity:\[\tan(75^{\circ}) = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\]Multiply numerator and denominator by the conjugate \( \sqrt{3} + 1 \) to simplify:\[\frac{(\sqrt{3} + 1)^2}{2} = \frac{3 + 2\sqrt{3} + 1}{2} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3}\]Thus, the exact value of \( \tan(75^{\circ}) \) is \( 2 + \sqrt{3} \).

Key Concepts

Mercator Projection

The Mercator Projection is a widely used method for creating a two-dimensional map of Earth. Named after Gerardus Mercator, a Flemish geographer, this projection was developed in 1569. It translates the globe into a flat, rectangular map.
This method is especially useful for navigation because it allows straight lines to represent constant compass bearings. However, as you move farther from the equator, the distances and sizes of landmasses become exaggerated. This is due to the way lines of latitude are spaced farther apart.

• At the equator, the spacing of latitude lines is normal.
• As latitude increases toward the poles, the distance between the lines increases.
• This distortion helps navigation but can misrepresent actual land sizes.

Overall, while Mercator Projection is essential for maritime charts, it is less suitable for accurate land representations in areas near the poles.

Latitude and Longitude

Latitude and longitude are two main components of geographic coordinates used to pinpoint any location on Earth. They form the crux of global location systems. Think of them like X and Y coordinates on a grid.
Latitude measures how far north or south a point is from the equator.

• It ranges from 0° at the Equator to 90° at the poles.
• Lines of latitude are called parallels; they run east-west.

Longitude measures how far east or west a point is from the Prime Meridian, which runs through Greenwich, England.

• It ranges from 0° at the Prime Meridian to 180° both east and west.
• Lines of longitude are called meridians; they run north-south.

Together, latitude and longitude form a grid covering the Earth, making it easier to locate places precisely, whether on a smartphone map or a paper atlas.

Tangent of Sum Formula

The Tangent of Sum Formula is a trigonometric identity used to calculate the tangent of the sum of two angles. This formula is especially useful when dealing with angles that aren’t standard.When you have two angles, say \( A \) and \( B \), the formula for the tangent of their sum is:\[\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}\]This formula breaks down into several key components:

• Addition in the numerator ties the tangents of the two angles.
• The denominator subtracts the product of the tangents of the angles.

In the exercise, we applied this formula to solve \( \tan(75^{\circ}) \) by setting \( A = 45^{\circ} \) and \( B = 30^{\circ} \).

• Knowing \( \tan(45^{\circ}) = 1 \) and \( \tan(30^{\circ}) = \frac{1}{\sqrt{3}} \), we plugged these values into the formula.
• After simplification, the exact result is \( 2 + \sqrt{3} \).

This technique is a cornerstone of trigonometry, simplifying complex angle calculations with relative ease.